Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins

Meiling Zhu; Huijun Xu; Meiling Zhu; Huijun Xu

doi:10.3934/mbe.2023297

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 6894-6911. doi: 10.3934/mbe.2023297

Previous Article Next Article

Research article

Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins

Meiling Zhu ,
Huijun Xu ^,

Yangzhou Polytechnic Institute, Yangzhou 225127, China

Academic Editor: Mariano Torrisi

Received: 16 November 2022 Revised: 19 January 2023 Accepted: 24 January 2023 Published: 07 February 2023

In this study, we investigate a delayed reaction-diffusion predator-prey system with the effect of toxins. We first investigate whether the internal equilibrium exists. We then provide certain requirements for the presence of Turing and Hopf bifurcations by examining the corresponding characteristic equation. We also study Turing-Hopf and Hopf bifurcations brought on by delays. Finally, numerical simulations that exemplify our theoretical findings are provided. The quantitatively obtained properties are in good agreement with the findings that the theory had predicted. The effects of toxins on the system are substantial, according to theoretical and numerical calculations.
- toxins,
- delay,
- diffusion,
- bifurcation,
- spatial pattern
Citation: Meiling Zhu, Huijun Xu. Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6894-6911. doi: 10.3934/mbe.2023297

Related Papers:

Abstract

In this study, we investigate a delayed reaction-diffusion predator-prey system with the effect of toxins. We first investigate whether the internal equilibrium exists. We then provide certain requirements for the presence of Turing and Hopf bifurcations by examining the corresponding characteristic equation. We also study Turing-Hopf and Hopf bifurcations brought on by delays. Finally, numerical simulations that exemplify our theoretical findings are provided. The quantitatively obtained properties are in good agreement with the findings that the theory had predicted. The effects of toxins on the system are substantial, according to theoretical and numerical calculations.

References

[1]	M. L. Crump, F. R. Hensley, K. L. Clark, Apparent decline of the golden toad: Underground or extinct?, Copeia, 2 (1992), 413–420. https://doi.org/10.2307/1446201 doi: 10.2307/1446201
[2]	T. G. Hallam, C. E. Clark, R. R. Lassiter, Effects of toxicants on populations: A qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Modell., 18 (1983), 291–304. https://doi.org/10.1016/0304-3800(83)90019-4 doi: 10.1016/0304-3800(83)90019-4
[3]	T. Das, R. N. Mukherjee, K. S. Chaudhuri, Harvesting of a prey–predator fishery in the presence of toxicity, Appl. Math. Modell., 33 (2009), 2282–2292. https://doi.org/10.1016/j.apm.2008.06.008 doi: 10.1016/j.apm.2008.06.008
[4]	R. Rani, S. Gakkhar, The impact of provision of additional food to predator in predator–prey model with combined harvesting in the presence of toxicity, J. Appl. Math. Comput., 60 (2019), 673–701. https://doi.org/10.1007/s12190-018-01232-z doi: 10.1007/s12190-018-01232-z
[5]	K. Chakraborty, K. Das, Modeling and analysis of a two-zooplankton one-phytoplankton system in the presence of toxicity, Appl. Math. Modell., 39 (2015), 1241–1265. https://doi.org/10.1016/j.apm.2014.08.004 doi: 10.1016/j.apm.2014.08.004
[6]	X. Wu, F. Wei, Single-species population models with stage structure and partial tolerance in polluted environments, Math. Biosci. Eng., 19 (2022), 9590–9611. https://doi.org/10.3934/mbe.2022446 doi: 10.3934/mbe.2022446
[7]	F. Wei, L. Chen, Psychological effect on single-species population models in a polluted environment, Math. Biosci., 290 (2017), 22–30. https://doi.org/10.1016/j.mbs.2017.05.011 doi: 10.1016/j.mbs.2017.05.011
[8]	F. Wei, S. A. H. Geritz, J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Lett., 63 (2017), 130–136. https://doi.org/10.1016/j.aml.2016.07.026 doi: 10.1016/j.aml.2016.07.026
[9]	X. Zhang, H. Zhao, Bifurcation and optimal harvesting of a diffusive predator–prey system with delays and interval biological parameters, J. Theor. Biol., 363 (2014), 390–403. https://doi.org/10.1016/j.jtbi.2014.08.031 doi: 10.1016/j.jtbi.2014.08.031
[10]	X. Zhang, H. Zhao, Dynamics analysis of a delayed reaction-diffusion predator-prey system with non-continuous threshold harvesting, Math. Biosci., 289 (2017), 130–141. https://doi.org/10.1016/j.mbs.2017.05.007 doi: 10.1016/j.mbs.2017.05.007
[11]	B. K. Das, D. Sahoo, G. Samanta, Impact of fear in a delay-induced predator–prey system with intraspecific competition within predator species, Math. Comput. Simul., 191 (2022), 134–156. https://doi.org/10.1016/j.matcom.2021.08.005 doi: 10.1016/j.matcom.2021.08.005
[12]	Y. Wang, X. Zou, On a predator–prey system with digestion delay and anti-predation strategy, J. Nonlinear Sci., 30 (2020), 1579–1605. https://doi.org/10.1007/s00332-020-09618-9 doi: 10.1007/s00332-020-09618-9
[13]	X. Jiang, X. Chen, T. Huang, H. Yan, Bifurcation and control for a predator-prey system with two delays, IEEE Trans. Circuits and Syst. Ⅱ: Express Briefs, 68 (2021), 376–380. https://doi.org/10.1109/TCSII.2020.2987392 doi: 10.1109/TCSII.2020.2987392
[14]	D. Duan, B. Niu, J. Wei, Hopf-hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos Solitons Fractals, 123 (2019), 206–216. https://doi.org/10.1016/j.chaos.2019.04.012 doi: 10.1016/j.chaos.2019.04.012
[15]	J. Liu, Y. Cai, J. Tan, Y. Chen, Dynamical behaviours of a delayed diffusive eco-epidemiological model with fear effect, Chaos, Solitons Fractals, 161 (2022), 112349. https://doi.org/10.1016/j.chaos.2022.112349 doi: 10.1016/j.chaos.2022.112349
[16]	D. Pal, G. P. Samanta, G. S. Mahapatra, Selective harvesting of two competing fish species in the presence of toxicity with time delay, Appl. Math. Comput., 313 (2017), 74–93. https://doi.org/10.1016/j.amc.2017.05.069 doi: 10.1016/j.amc.2017.05.069
[17]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, Springer, New York, 1996.
[18]	H. Zhu, X. Zhang, G. Wang, L. Wang, Effect of toxicant on the dynamics of a delayed diffusive predator-prey model, J. Appl. Math. Comput., 69 (2023), 355–379. https://doi.org/10.1007/s12190-022-01744-9 doi: 10.1007/s12190-022-01744-9
[19]	X. Yan, Y. Li, G. Guo, Qualitative analysis on a diffusive predator-prey model with toxins, J. Math. Anal. Appl., 486 (2020), 123868. https://doi.org/10.1016/j.jmaa.2020.123868 doi: 10.1016/j.jmaa.2020.123868
[20]	G. P. Samanta, A two-species competitive system under the influence of toxic substances, Appl. Math. Comput., 216 (2010), 291–299. https://doi.org/10.1016/j.amc.2010.01.061 doi: 10.1016/j.amc.2010.01.061
[21]	X. Zhang, Q. An, L. Wang, Spatiotemporal dynamics of a delayed diffusive ratio-dependent predator–prey model with fear effect, Nonlinear Dyn., 105 (2021), 3775–3790. https://doi.org/10.1007/s11071-021-06780-x doi: 10.1007/s11071-021-06780-x
[22]	W. Ni, M. Wang, Dynamics and patterns of a diffusive Leslie–Gower prey–predator model with strong Allee effect in prey, J. Differ. Equations, 261 (2016), 4244–4274. https://doi.org/10.1016/j.jde.2016.06.022 doi: 10.1016/j.jde.2016.06.022
[23]	W. Zuo, J. Wei, Stability and hopf bifurcation in a diffusive predator–prey system with delay effect, Nonlinear Anal.: Real World Appl., 12 (2011), 1998–2011. https://doi.org/10.1016/j.nonrwa.2010.12.016 doi: 10.1016/j.nonrwa.2010.12.016
[24]	S. Chen, J. Shi, J. Wei, Time delay-induced instabilities and hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1–38. https://doi.org/10.1007/s00332-012-9138-1 doi: 10.1007/s00332-012-9138-1
[25]	S. Chen, J. Shi, J. Wei, Global stability and hopf bifurcation in a delayed diffusive Leslie–Gower predator-prey system, Int. J. Bifurcation Chaos, 22 (2012), 1250061. https://doi.org/10.1142/S0218127412500617 doi: 10.1142/S0218127412500617
[26]	L. A. Howland, A solution of the biquadratic equation, Am. Math. Mon., 18 (1911), 102–108. https://doi.org/10.1080/00029890.1911.11997617 doi: 10.1080/00029890.1911.11997617
[27]	P. Y. Pang, M. Wang, Qualitative analysis of a ratio-dependent predator–prey system with diffusion, Proc. R. Soc. Edinburgh Sect. A: Math., 133 (2003), 919–942. https://doi.org/10.1017/S0308210500002742 doi: 10.1017/S0308210500002742

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)